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3 pp. Presented at several mathematical conferences.
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Non-commutativity, Dirac, quantum mechanics
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\begin{document}
\title{A Note on the Dirac-like Equation in Non-commutative Geometry}
\author{Valeriy V. Dvoeglazov\\
Universidad de Zacatecas\\
Apartado Postal 636, Suc. 3\\
Zacatecas 98061, Zac., M\'exico\\
E-mail: valeri@fisica.uaz.edu.mx}

%\date{June 17, 2003}
\date{\empty}

\maketitle

\begin{abstract}
We postulate the non-commutativity of 4-momenta and we derive the mass splitting in the Dirac equation.
The applications are discussed.
\end{abstract}

The non-commutativity~\cite{snyder,amelino} manifests interesting peculiarities in
the Dirac case.
Recently, we analized Sakurai-van der Waerden method of derivations of the Dirac
(and higher-spins too) equation~\cite{Dvoh}. We can start from
\begin{equation}
(E I^{(2)}-{\bm \sigma}\cdot {\bf p}) (E I^{(2)}+ {\bm\sigma}\cdot
{\bf p} ) \Psi_{(2)} = m^2 \Psi_{(2)} \,,
\end{equation}
or
\begin{equation}
(E I^{(4)}+{\bm \alpha}\cdot {\bf p} +m\beta) (E I^{(4)}-{\bm\alpha}\cdot
{\bf p} -m\beta ) \Psi_{(4)} =0.\label{f4}
\end{equation}
As in the original Dirac work, we have
\begin{equation}
\beta^2 = 1\,,\quad
\alpha^i \beta +\beta \alpha^i =0\,,\quad
\alpha^i \alpha^j +\alpha^j \alpha^i =2\delta^{ij} \,.
\end{equation}
For instance, their explicite forms can be chosen 
\begin{eqnarray}
\alpha^i =\pmatrix{\sigma^i& 0\cr
0&-\sigma^i\cr}\,,\quad
\beta = \pmatrix{0&1_{2\times 2}\cr
1_{2\times 2} &0\cr}\,,
\end{eqnarray}
where $\sigma^i$ are the ordinary Pauli $2\times 2$ matrices.

We also postulate the non-commutativity relations for the components of 4-momenta:
\begin{equation}
[E, {\bf p}^i]_- = \Theta^{0i} = \theta^i\,,
\end{equation}
as usual. Therefore the equation (\ref{f4}) will {\it not} lead
to the well-known equation $E^2 -{\bf p}^2 = m^2$. Instead, we have
\begin{equation}
\left \{ E^2 - E ({\bm \alpha} \cdot {\bf p})
+({\bm \alpha} \cdot {\bf p}) E - {\bf p}^2 - m^2 - i ({\bm\sigma}\otimes I_{(2)})
[{\bf p}\times {\bf p}] \right \} \Psi_{(4)} = 0
\end{equation}
For the sake of simplicity, we may assume the last term to be zero. Thus, we come to
\begin{equation}
\left \{ E^2 - {\bf p}^2 - m^2 -  ({\bm \alpha}\cdot {\bm \theta})
\right \} \Psi_{(4)} = 0\,.
\end{equation} 
However, let us apply the unitary transformation. It is known~\cite{Berg,Dvoe}
that one can\footnote{Some relations for the components ${\bf a}$ should be assumed. Moreover, in our case ${\bm \theta}$ 
should not depend on $E$ and ${\bf p}$. Otherwise, we must take the non-commutativity $[E, {\bf p}^i]_-$ into account again.}
\begin{equation}
U_1 ({\bm \sigma}\cdot {\bf a}) U_1^{-1} = \sigma_3 \vert {\bf a} \vert\,.\label{s3}
\end{equation}
For ${\bm \alpha}$ matrices we re-write (\ref{s3}) to
\begin{eqnarray}
{\cal U}_1 ({\bm \alpha}\cdot {\bm \theta}) {\cal U}_1^{-1} = \vert {\bm \theta} \vert
\pmatrix{1&0&0&0\cr
0&-1&0&0\cr
0&0&-1&0\cr
0&0&0&1\cr} = \alpha_3 \vert {\bm\theta}\vert\,.
\end{eqnarray}
The explicit form of the $U_1$ matrix is ($a_{r,l}= a_1\pm ia_2$):
\begin{eqnarray}
U_1 &=&\frac{1}{\sqrt{2a (a+a_3)}} \pmatrix{a+a_3&a_l\cr
-a_r&a+a_3\cr}  = \frac{1}{\sqrt{2a (a+a_3)}} [ a+a_3 + i\sigma_2 a_1 - i\sigma_1 a_2]\,,\nonumber\\ 
{\cal U}_1 &=&\pmatrix{U_1 &0\cr
0& U_1 \cr}\,.
\end{eqnarray}
Let us apply the second unitary transformation:
\begin{eqnarray}
{\cal U}_2 \alpha_3 {\cal U}_2^\dagger =
\pmatrix{1&0&0&0\cr
0&0&0&1\cr
0&0&1&0\cr
0&1&0&0\cr} \alpha_3 \pmatrix{1&0&0&0\cr
0&0&0&1\cr
0&0&1&0\cr
0&1&0&0\cr} = \pmatrix{1&0&0&0\cr
0&1&0&0\cr
0&0&-1&0\cr
0&0&0&-1\cr}\,.
\end{eqnarray}
The final equation is
\begin{equation}
[E^2 -{\bf p}^2 -m^2 -\gamma^5_{chiral} \vert {\bm \theta}\vert ] \Psi^\prime_{(4)} = 0\,.
\end{equation}
In the physical sense this implies the mass splitting for a Dirac particle over the non-commutative space, $m_{1,2} =\pm \sqrt{m^2 \pm \theta}$. 
This procedure may be attractive for explanation of the mass creation and the mass splitting for fermions.


\begin{thebibliography}{99}

\footnotesize{
\bibitem{snyder}  H. Snyder, Phys. Rev. {\bf 71}, 38 (1947); ibid. {\bf 72},
68 (1947).

\bibitem{amelino}  A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D{\bf 52}
, 1108 (1995); G. Amelino-Camelia, Nature {\bf 408}, 661 (2000);
gr-qc/0012051; hep-th/0012238; gr-qc/0106004; J. Kowalski-Glikman,
hep-th/0102098; G. Amelino-Camelia and M. Arzano, hep-th/0105120; N. R.
Bruno, G. Amelino-Camelia and J. Kowalski-Glikman, hep-th/0107039.

\bibitem{Dvoh} V. V. Dvoeglazov, Rev. Mex. Fis. Supl. {\bf 49},  99 (2003) ({\it Proceedings of the DGFM-SMF School, Huatulco, 2000})

\bibitem{Berg} R. A. Berg, Nuovo Cimento {\bf 42}A, 148 (1966). 

\bibitem{Dvoe} V. V. Dvoeglazov, Nuovo Cimento A{\bf 108}, 1467 (1995).

}

\end{thebibliography}



\end{document}
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